for part a . please STRICTLY follow the hint provided in part a . express y2(t) in terms of y1, p , AND q. express it as a deinfite integral. then simplify it by removing complicated constant factors and leave out any multiples of y1.please ALSO answer the quesiton in part a "under which assumption is the resulting formula for y2 valid. "this is NOT. a graded quesitonthere is no other information needed. all of the information are provided. 1.solution ofSuppose the functions p(t) and q(t) are continuous on the interval I = (a,b). Let y1(t) be ay" + p(t)y' + q(t)y = 0(1)that has no zeroes on (a, b), and let to € (a, b).(a)(b)Assuming y₁(t) is a given solution to the ODE (1), derive a formula for any otherlinearly independent solution y2(t) for (1) in terms of the solution y₁ and the coefficients p, q.(Hint: You may express the solution y2(t) as a definite integral between to and t of the other givenfunctions. You may simplify your final formula by (i) leaving out any additive terms that aremultiples of y1(t), and (ii) removing any complicated constant factors.)Under which assumptions is this formula valid?Check that the given y₁ together with the y2 derived in part (a) form a fundamental setof solutions for (1).

(a)Given the 2nd order DE , where p(t) and q(t) are continuous on the interval I=(a,b). Let y1(t) be…

solution of Suppose the functions p(t) and q(t) are continuous on the interval I = (a,b). Let y₁(t) be a y"+p(t)y'+q(t)y = 0 that has no zeroes on (a, b), and let to € (a, b). (a) (b) (1) Assuming y₁(t) is a given solution to the ODE (1), derive a formula for any other linearly independent solution y2(t) for (1) in terms of the solution y₁ and the coefficients p, q. (Hint: You may express the solution y2(t) as a definite integral between to and t of the other given functions. You may simplify your final formula by (i) leaving out any additive terms that are multiples of y1(t), and (ii) removing any complicated constant factors.) Under which assumptions is this formula valid? Check that the given y₁ together with the y2 derived in part (a) form a fundamental set of solutions for (1).

solution of Suppose the functions p(t) and q(t) are continuous on the interval I = (a,b). Let y₁(t) be a y"+p(t)y'+q(t)y = 0 that has no zeroes on (a, b), and let to € (a, b). (a) (b) (1) Assuming y₁(t) is a given solution to the ODE (1), derive a formula for any other linearly independent solution y2(t) for (1) in terms of the solution y₁ and the coefficients p, q. (Hint: You may express the solution y2(t) as a definite integral between to and t of the other given functions. You may simplify your final formula by (i) leaving out any additive terms that are multiples of y1(t), and (ii) removing any complicated constant factors.) Under which assumptions is this formula valid? Check that the given y₁ together with the y2 derived in part (a) form a fundamental set of solutions for (1).

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter14: Discrete Dynamical Systems

Section14.3: Determining Stability

Problem 13E: Repeat the instruction of Exercise 11 for the function. f(x)=x3+x For part d, use i. a1=0.1 ii...

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